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We consider the inverse problem of identifying parameters in a variant of thediffuse interface model for tumour growth model proposed by Garcke, Lam, Sitkaand Styles (Math. Models Methods Appl. Sci. 2016). The model contains threeconstant parameters; namely the tumour growth rate, the chemotaxis parameterand the nutrient consumption rate. We study the inverse problem from theviewpoint of PDE-constrained optimal control theory and establish first orderoptimality conditions. A chief difficulty in the theoretical analysis lies inproving high order continuous dependence of the strong solutions on theparameters, in order to show the solution map is continuously Fr\'{e}chetdifferentiable when the model has a variable mobility. Due to technicalrestrictions, our results hold only in two dimensions for sufficiently smoothdomains. Analogous results for polygonal domains are also shown for the case ofconstant mobilities. Finally, we propose a discrete scheme for the numericalsimulation of the tumour model and solve the inverse problem using atrust-region Gauss--Newton approach.

Parameter identification via optimal control for a Cahn--Hilliard-chemotaxis system with a variable mobility